Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model
For practitioners using Monte Carlo simulation of stochastic volatility models, this work identifies and characterizes numerical instabilities arising from discretization, which is an incremental contribution.
The paper analyzes the Euler-Maruyama discretization of the Hull-White stochastic volatility model, showing that the discretized model exhibits a phase transition in Lyapunov exponents of asset price moments, leading to numerical explosions. It provides explicit asymptotic limits, fluctuation results, and criteria for these explosions.
We consider the stochastic volatility model $dS_t = σ_t S_t dW_t,dσ_t = ωσ_t dZ_t$, with $(W_t,Z_t)$ uncorrelated standard Brownian motions. This is a special case of the Hull-White and the $β=1$ (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the $n\to \infty$ limit of a very large number of time steps of size $τ$, at fixed $β=\frac12ω^2τn^2$ and $ρ=σ_0^2τ$, and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of $S_t$. Under the Euler-Maruyama discretization for $(S_t,\log σ_t)$, the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.